simulation of free-surface flows with...
TRANSCRIPT
Simulation of Free-Surface Flows
With STAR-CCM+
Samir Muzaferija and Milovan Perić
CD-adapco
Contents
Introduction to multiphase flows
Theoretical background for VOF-method
High-Resolution Interface-Capturing (HRIC) scheme
Accounting for surface tension effects
Extensions of VOF-method
Waves: generation and propagation
Free surface flows: application examples
Future development
Introduction to Multiphase Flows
VOF-approach is suitable, when
the grid is fine enough to resolve
the interface between two
immiscible fluids.
Sometimes not all parts of the flow
are suited for VOF-treatment…
Examples: Atomization nozzle
flow and jet break-up (right) and
flow around a hydrofoil (below)
Interface Conditions
• Conditions at an interface between two immiscibe fluids:
Kinematic condition: No flow through interface.
Dynamic conditions: Balance of normal and tangential stresses (surface tension forces):
VOF: Theory, I
• VOF considers a single effective fluid whose properties vary according to volume fraction of individual fluids:
• The mass conservation equation for fluid i reads:
• It can be rearranged into an equation in integral form:
This equation is used to compute the transport of volume fraction αi.
VOF: Theory, II
• The mass conservation equation for the effective fluid is obtained by summing up all component equations and using the condition:
• The integral form of mass conservation equation (used to compute pressure correction) reads:
• The properties of effective fluid are computed according to volume fractions:
Interface-Capturing Method, I
• For sharp interfaces, special discretization for convective terms in the equation for volume fraction αi is needed (to avoid excessive spreading).
• The method must produce bounded solutions, i.e. each volume fraction must lie between 0 and 1 and the sum of all volume fractions must be 1 at each control volume.
• Bounded schemes must fall within a certain region of the normalized variable diagram; the normalized variables are defined as:
Interface-Capturing Method, II
• The boundedness requirement:
The normalized variable
diagram and the proposed
high-resolution interface-
capturing (HRIC) scheme
(details available in STAR-
CCM+ documentation)
HRIC-Scheme, IV
Simulation of sloshing in a tank due to sinusoidal sway motion:
one-cell sharp interface before wave overturns (left) and smeared
Interface after splashing (right), when the interface is in reality not sharp…
Interface Sharpening
• In order to prevent dilution, one can activate “interface sharpening” by setting “Sharpening factor” to a value >0.
• The sharpening model is based on “anti-diffusion” and acts only in cells at the interface…
• This is usually required only for violent sloshing and similar phenomena…
Local Grid Refinement, I
• One should, when possible, align grid with free surface where it is flat…
• One should, when possible, avoid vertical grid coarsening in free-surface zone where its deformation is small…
• The reason: volume fraction is convected into finer cells and leads to smeared interface…
Flow around a vertical cylinder – two grids for the same initial free surface position
Local Grid Refinement, II
Initial value from this cell feeds into next two, from there into next four – the smeared
interface does not get sharper by refining time step (only “Sharpening Factor” helps –
but it is better to adapt the grid to free surface that to use artificial anti-diffusion…)
Impulsively started flow around a vertical cylinder
Surface Tension Effects, I
• The kinematic interface condition is implicitly accounted for by the transport equation for volume fraction.
• The dynamic interface conditions require additional forces in the momentum equations in cells containing free surface…
• Surface tension forces are converted to volume forces:
Since the gradient of volume
fraction is zero away from
interface, these terms are
equal to zero everywhere
except along interface…
Surface Tension Effects, II
• The unit vector normal to interface is obtained from the gradient of volume fraction:
• The curvature of free surface is obtained from the divergence of the unit vector normal to interface:
• The volume fraction field needs to be smoothed before the curvature is computed (sharp interface leads to a non-smooth curvature field).
Surface Tension Effects, III
• The so called „parasitic currents“ can develop, if the fluid moves only slowly or not at all, and the surface tension effects dominate (high curvature or surface tension coefficient)...
• The reason: pressure and surface tension forces must be in equilibrium when fluid is at rest – but the numerical approximations do not guarantee that (one term is linear and the other is non-linear):
• There are many partial solutions to this problem in literature, but none works in all situations…
Surface Tension Effects, IV
• Recently, a new model called “Interface Momentum Dissipation” was introduced in STAR-CCM+ to reduce the effects of parasitic currents…
• The momentum dissipation term is added to the momentum equations only in the vicinity of the interface…
• It acts similarly as an increased fluid viscosity near interface (more on the gas side): µint grad(v)
• Interface Momentum Dissipation decreases rapidly with distance from interface…
Surface Tension Effects, V
• Where free surface is in contact with wall, contact angle needs to be prescribed.
Surface Tension Effects, VI
• One can distinguish between:
Static contact angle
Dynamic advancing contact angle on dry surface
Dynamic advancing contact angle on wet surface
Dynamic receding contact angle
• The contact angle is enforced as:
nfs = - n
w cos θ
w + t
w sin θ
w
Interface Momentum Dissipation:
Ink Jet Droplet, I
Without IMD
With IMD
Without IMD, parasitic currents are strong (maximum velocity 35.88 m/s);
With IMD, parasitic currents are hardly visible (maximum velocity 8.98 m/s)
Without IMD, the interface is smeared behind secondary droplet and at nozzle exit;
With IMD, the interface is sharp almost everywhere…
Without IMD
With IMD
Interface Momentum Dissipation:
Ink Jet Droplet, II
Without IMD:
Strong parasitic
currents, maximum
velocity 4.97 m/s
(10x web speed)
With IMD:
Very weak parasitic
currents, maximum
velocity 0.506 m/s (1%
above web speed)
Interface Momentum Dissipation:
Flow in a Slot Coater, I
Interface Momentum Dissipation:
Flow in a Slot Coater, II
Without IMD:
Front meniscus has
irregular shape due
to high parasitic
velocities
With IMD:
Smooth front
meniscus
Interface Momentum Dissipation:
Flow in a Slot Coater, II
Without IMD:
Flow rate at outlet fluctuates due to
high parasitic velocities
With IMD:
Flow rate at outlet fluctuates less
Interface Momentum Dissipation:
Flow in and Around a Rising Bubble
Left: Without IMD
Strong parasitic currents, maximum
velocity 11.68 m/s, interface smeared
through high velocity normal to it, the
flow inside bubble cannot be
recognized…
Right: With IMD
Hardly visible parasitic currents,
maximum velocity 0.39 m/s (30
times lower than before), interface is
sharp (resolved by one cell) and one
can clearly see the flow inside
bubble…
Extensions of VOF-Method
• One can add additional models in the equation for volume fraction (diffusion, sources) in order to model effects like non-sharp interfaces, phase change etc.
• This is the main advantage of this approach compared to level-set and similar schemes...
• VOF-framework is already used in STAR-CCM+ for the following models:
Evaporation and condensation
Melting and solidification
Cavitation
Boiling
• STAR-CCM+ provides several wave models:
– For initialization of volume fraction, velocity and pressure fields;
– For transient inlet boundary conditions.
• Currently available models:
– 1st-order linear wave theory
– Non-linear 5th-order Stokes wave theory (Fenton, 1985)
– Pierson-Moskowitz and JONSWAP long-crested wave spectra
– Superposition of linear waves with varying amplitude, period and direction of propagation (can be set-up via Excel-file)
Wave Models
• Accurate wave propagation requires 2nd-order time-integration
method.
• Second-order method (quadratic interpolation in time) requires
that the wave propagates less than half a cell per time step.
• First-order scheme is always stable but less accurate…
Time-Accurate Wave Propagation
Scaled 10 times in vertical direction…
Stokes 5th-order wave after 11 periods (8.977 s), resolved by 80 cells per wave-
length (125 m) and 20 cells per wave height (5 m); damping over the last 300 m
Internal Wave Generation
• The source term in equation for volume fraction can be used to simulate injection and suction…
• … which can be used to create waves at free surface…
• By a suitable choice of the position and shape of the “source zone” and an appropriate source term function, one can generate waves of desired shape…
• The advantage of this approach: waves radiated by a solid structure can pass over the source region without reflection (which happens when waves are created by inlet boundary conditions)
• Improvements to the treatment of contact angle (better
recognition of contact line, distinguishing direction of
motion etc.)
• Transition to other multiphase models:
– VOF to Lagrangian and vice-versa
– Fluid film to VOF and vice versa
• Eulerian or Lagrangian multiphase models within VOF
phases
Future Developments: VoF
Simulation of Pouring